Question

y=sinx,y=1/2,y=(√2)/2所围成图形的面积，求详细过程

是右曲线减左曲线么？他们的函数有啥差别？

Answer 1

如图，y=sinx,y=1/2,y=(√2)/2 在 [0,2π] 内所围成图形的面积是图中类似曲边梯形的红色部分，可用 y=sinx, y=1/2 围成的大帽子减去 y=sinx, y=(√2)/2 围成的小帽子求出。y=sinx, y=1/2 的交点横坐标是 x= π/6, 5π/6；y=sinx, y=(√2)/2 的交点横坐标是 x= π/4, 3π/4

[π/6, 5π/6]∫(sinx-1/2)dx-[π/4, 3π/4]∫(sinx-√2/2)dx

=(-cosx-0.5x)|[π/6, 5π/6]-(-cosx-x√2/2)|[π/4, 3π/4]

=-cos(5π/6)-5π/12+cos(π/6)+π/12+cos(3π/4)+(3√2)π/8-cos(π/4)-(√2)π/8

=√3-π/3-√2+(√2)π/4

≈0.381360429